Buffering effect of fiber coating and its influence on the prooftest load in optical fibers

Ephraim Suhir

The tensile force experienced by a coated glass fiber during proof testing is applied to the coating and,therefore, may be reduced because of the buffering effect of the coating. In this study such an effect isevaluated on the basis of an analytical stress model. It is shown that for any finite, even very great, coatingcompliance, one can always choose a sufficiently long specimen in which the glass fiber is loaded to practicallythe same level as in an infinitely long composite. The numerical example carried out for a dual-coated fiber,considered for future undersea systems, indicated that if -10 cm long specimens are employed, 98% of thefiber length will be subjected to a tensile force >99% of the value for the case of an infinitely long fiber. Theresults obtained can be used for the analysis of the mechanical performance of coated fibers of finite length.They can be utilized, in particular, for selecting appropriate specimen lengths during proof testing.

1. IntroductionOptical fibers employed in lightguide cables are

proof tested to a given tensile stress before cabling.'An appropriate proof test force must be chosen toensure that the fibers are able to withstand cable oper-ations such as laying, recovery and holding, and tosurvive for a long as twenty five years in the ocean.2Since the tensile force acting on the glass fiber is ap-plied to the coating, it can be affected by the mechani-cal properties and thickness(es) of the coating layer(s).It has been shown experimentally3 that in single-coat-ed optical fibers the polyurethane acrylate coating canshare an essential fraction of the tensile load impactedto the glass-coating composite. Intuitively it is feltthat this fraction could be even greater in dual-coatedfibers with soft primary coatings.

In the analysis below, we evaluate the buffering ef-fect of fiber coating and its effect on the proof test forcetransmitted to the glass. The study is performed inapplication to recently developed "hermetic" fibers,4which are thought to be likely candidates for futureundersea systems.5 For comparison, we evaluate thebuffering effect of coating in single-coated fibers aswell.

The author is with AT&T Bell Laboratories, Murray Hill, NewJersey 07974.

Received 28 August 1989.0003-6935/90/182682-04$02.00/0.© 1990 Optical Society of America.

II. AnalysisOur analysis is based on the following major assump-

tions:Although most low module materials used as prima-

ry coatings are, in effect, viscoelastic solids, in thisstudy we do not consider the viscoelastic properties ofthese materials and treat them as linearly elastic sol-ids. We believe that this simplification is acceptable,as long as the purpose of our analysis is to assess theattenuation length for the stresses applied at the endpositions of a specimen, rather than a more or lessaccurate prediction of the stress level.

No tensile or compressive forces occur in the prima-ry coating. This is subjected to shear only. Such anassumption is justified as long as Young's modulusand/or the thickness of the primary coating material issignificantly smaller than Young's moduli and cross-sectional dimensions of the glass fiber and the second-ary coating.6 In the case in question, this assumptionmeans that the shearing stress ro at the interface be-tween the glass and the primary coating and the shear-ing stress rT at the coatings interface are inverselyproportional to the corresponding radii (Fig. 1):

To(X) r,

rl(x) ro (1)

The above stresses can be determined from the fol-lowing condition of the compatibility for the longitudi-nal (axial) interfacial displacements:

UO(X) = U2 (X) - Klro(X). (2)

Here, uo and u2 are the displacements of the pointslocated on the surface of the glass fiber and on theinner surface of the secondary coating, respectively,

2682 APPLIED OPTICS / Vol. 29, No. 18 / 20 June 1990

_4 _. . . . .X. u s x

P ArA\E4eX-x\XN~xE* z1 P+_ I~~~~~~

GLASS PRIMARY SECONDARYFIBER COATING COATING

Fig. 1. Dual coated fiber specimen.

= XO + 2, K = KO + K + K2 - (8)

are the total axial and the total interfacial compliancesof the composite, respectively. By differentiating Eq.(7) we have:

KT7(X) - XTO(x) = 2P

The next differentiation yields:

T;(X) - K 2 To(x) = 0,

(9)

,,, l (10)

and Kl is the compliance of the primary coating. If thestresses ro(x) and ri(x) were known, then the displace-ments u0 and U2 could be evaluated as

UO = X0 Jo TO(Q)dt + Koro(x)

U2 = X2 J To()dt - K2 rl(X) + K2 PX

where

To(x) = 27rro Jf To()dt (4)

is the axial force in the glass fiber and in the secondarycoating,

1 A 12 1irr 0E0 2 7r(r2 - r)E 2 (5)

are axial compliances of these components, and E0 andE2 are Young's moduli of the materials. The firstterms in Eq. (3) are due to the axial forces To and areevaluated in accordance with Hooke's law assumingthat these forces are uniformly distributed over thecross-sectional areas. The second terms account forthe nonuniform radial distribution of the forces To andreflect an assumption that the corresponding correc-tions depend only on the stress in the given crosssection and are not affected by the stresses acting inother cross sections. The third term in the second Eq.(3) is due to the external load P. The interfacialcompliances KO, Ki and K2 can be evaluated by the for-mulas:

r0 r 0 ro

(1 + 3&y)(1 - y) + 7(1 + V2) ln 1 + r, ~~~~~~~~~~2-yK2 =-

where -y = rlr2. These formulas can be obtained onthe basis of the solution to the theory of elasticity 7

problem for a hollow cylinder loaded by uniformlydistributed axial shearing forces.8

Introducing Eq. (3) into Eq. (2) and considering Eq.(1), we obtain the following integral equation for theshearing stress function To(x):

KTO(X) - A ( TOQ)dt = A2Px, (7)

where

where

K = 2-rro-K

is the eigenvalue of the problem.Equation (10) has the following solution:

To(X) = CO sinhkx + Cl coshkx.

(11)

(12)

Since the function ro(x) must be antisymmetric withrespect to the origin, and cosh kx is a symmetric func-tion, then one must put C = 0. The constant Co canbe determined from the boundary condition

Tr'(e) = A2 p. (13)

This condition can be obtained from Eq. (9), keeping inmind that no external forces are applied at the ends ofthe glass fiber and, therefore, To(e) = 0. Then wehave:

CO= X2p° KK coshke '

and the shearing stress rO(x) is expressed as

sinhkxTO(X) = Tmax sinhke '

where

Imax = kP tanhke = k -2 p tanhkekK X~~

(14)

(15)

(16)

is the maximum value of this stress. In sufficientlylong (large e values) fibers with stiff (large k values)interfaces, the maximum shearing stress becomes fiberlength independent (since tanhke - 1 when ke -> o)and Eq. (15) can be presented as follows:

To(x) = Tmax exp[-k(e - x)]. (17)

This formula indicates that the stress ro(x) is verysmall in the midportion of the specimen (where x isessentially smaller than 2) and concentrates near thespecimen's ends (where x is on the same order of mag-nitude as e).

Introducing Eq. (15) into Eq. (4), we obtain thefollowing formula for the force in the glass fiber:

(18)TO(X) =-7PX(X),

where the factor10= 1 , OSq < 1,

fAA2

(19)

20 June 1990 / Vol. 29, No. 18 / APPLIED OPTICS 2683

FI~ I

I - 'y2

I

-, kb)

determines the fraction of the proof test load P tramitted to an infinitely long glass fiber, and the funct

coshkxX( ) oshke ' <x (

characterizes the longitudinal distribution of theforces To(x) and the effect of the finite length of thespecimen. This function is maximum in the mid-crosssection x = 0:

Xmax = 1 - hke (21)

For sufficiently large k, the factor Xmax is close tounity. In this case, the fraction of the proof test forceP experienced by the fiber becomes independent ofwhether this force is applied to the entire cross sectionof the composite or to the secondary coating only.Indeed, if the external force is applied to the entirecross section, the forces acting on the glass and thesecondary coating are inversely proportional to theaxial compliances of these components, i.e.,

To EoAoP To E2A2

or

X7 Eoro X2

1 1E2(r2 - r2) Xo

Solving this equation for X we obtain Eq. (19).For long specimens with stiff interfaces, the Eq. (20)

can be simplified as follows:

x(x) = 1 - exp[-k(l - x)]. (22)

As evident from this formula, the factor x is close tounity for the cross sections x sufficiently remote fromthe specimen's ends (x << ), so that the force Toexperienced by the glass is practically constant alongthe fiber. Only near the ends, when the coordinate xbecomes close to e, the force To exponentially dropsand turns to zero at the end planes.

Obviously, for any finite, even very great, interfacialcompliance of the coating(s), resulting in small k val-ues, one can always choose such a fiber length L = 2ethat would lead to a sufficiently large force within therequired (or desired) portion of the glass fiber. Thislength can be evaluated by the equation:

L = 2e = 2 ,n(l X) (23)k x

e

which follows from Eq. (20).Note that the developed theory is equally applicable

to single-coated fibers. In this case, the compliance Klof the primary coating should be simply put equal tozero.

Ill. Numerical Examples

(1) The calculated data for a coated glass fiber (ro =62.5 Am, E0 = 7.19 X 1010 Pa = 10.4 X 106 psi; rl = 92.0Am, El = 6.89 X 105 Pa = 100 psi, v, = 0.5, r2 = 127.5um, E2 = 1.58 X 107 Pa = 2285 psi, v2 = 0.35) is:

X = 1.1333 X 10-3 NI X2 = 1.3754 N. X = 1.3765 1

(20) K0 = 8.693 X 106 N. K = 1.052 X 10 N

K2 = 4.502 X 10- m K = 2220-.

If we require, for instance, that 99% of the speci-men's length be subjected to a tensile load that is notsmaller than 99% of the load which the glass fiberwould experience in an infinitely long composite, thenthe length of the specimen, in accordance with Eq.(23), should not be smaller than

L=2.e 2 n(1- 0.99) =044m2220 1 - 0.99 =0.414 m-

On the other hand, if one accepts that only the mid-cross section of the specimen experiences a load equalto 99% of the level in an infinitely long composite, then,assuming x = 0, we find L = 0.00207 m 2 mm. Notethat the fraction of the external force experienced bythe glass approaches that for an infinitely long com-posite and, in accordance with Eq. (19), = 0.999, i.e.,almost the entire load is transmitted to the glass.

(2) In the test conducted by Krause et al.,3 single-coated specimens with the following characteristicswere used: r = 64 ,im, r = 1.34 ,im, E = 7.03 X1010N/m2, El = 5.65 X 108N/m2.9 In these specimensX = 1.105 X 10-3 (1/N), X1 = 0.04065(1/N), and, there-fore, q = 0.9735. This is in fairly good agreement withthe experimental results of Krause et al.3 The calcu-lated interfacial compliances for the materials used inthese experiments are Ko = 9.104 X 10'16(m3/N), K =4.571 X 10-1 3 (m3/N), K = 4.580 X 10-13(m3/N), and theeigenvalue k = 301940(1/m). With such a great kvalue, 99% of the fiber length will experience a tensileforce which is not smaller than 99% of the load in thecase of an infinitely long fiber, if the length of thespecimen is not smaller than only 3 mm.

IV. ConclusionsThe following major conclusions can be drawn from

the executed analysis:The fiber coating may share an essential fraction of

the tensile load imparted to the coated fiber duringproof testing and, therefore, this effect should be ac-counted for when evaluating the load experienced bythe glass fiber.

In infinitely long fibers, the above fraction can bedetermined assuming that the forces experienced bythe glass fiber and the (secondary) coating are inverse-ly proportional to the axial compliances of the glassand the coating. This fraction is independent ofwhether the external load is applied to the entire crosssection of the composite or to the coating only. Thiseffect may be neglected also in fibers of finite length,i.e., for actual specimens, if the external load is appliedto the entire cross section of the composite. Such asituation can take place, for instance, if in addition to

2684 APPLIED OPTICS / Vol. 29, No. 18 / 20 June 1990

the tensile force, a sufficiently large lateral pressure isapplied to the ends of the structure.

In fibers of finite length subjected to proof test loadapplied to the coating, the fraction of this load experi-enced by the glass may be essentially smaller than in aninfinitely long fiber. In addition, the forces acting inthe cross sections of the glass and the coating of fibersof finite length are no longer uniform along the fiber.The force in the glass is the largest in its mid-portionand drops to zero at the end. The force acting in thecoating is, on the contrary, the largest at the speci-men's ends and is very small in its mid-portion. Themagnitude and the distribution of the force in the glasscan be determined by multiplying the force calculatedfor an infinitely long fiber by the factor x (x). Thefiber length 2 and the compliance factor k play asimilar role as far as the maximum value of this factor xis concerned. Therefore, for any finite, even verylarge, interfacial compliance of the coating, resulting insmall k values, one can always choose such a length 2eof the specimen, that the force in the glass fiber besufficiently high within the required (or desired) por-tion of the glass fiber.

Calculations carried out for a dual-coated fiber indi-cate that if one requires that 99% of the length of theglass fiber experiences a proof test load not smallerthan 99% of the value in the case of an infinitely longfiber (when the buffering effect of the coating is of noimportance), a 414 mm long specimen should be em-ployed. In 104 mm long specimens, 98% of theirlengths experience forces of the above level. In 41.5

mm long specimens, only 95% of their lengths are sub-jected to such a load.

The author is grateful to L. L. Blyler, Jr., R. Tumin-aro, J. T. Krause, G. M. Bubel, and G. A. Decker foruseful discussions.

References1. S. E. Miller and A. G. Chynoweth, Eds. Optical Fiber Telecom-

munications (Academic, New York, 1979).2. T. C. Chu and H. C. Chandan, "Determination of Fiber Proof-

Test Stress for Undersea Lightguide Cable," AT&T Tech. J. 64,971-982 (1985).

3. J. T. Krause, C. R. Kurkjian, F. D. DiMarcello, and R. G. Huff,"Mechanical Reliability of Hermetic Carbon Coated Optical Fi-bers," in Proceedings, Sixth EFOC/LAN Conference Amster-dam, 121-123 (1988).

4. R. G. Huff, F. V. DiMarcello, and A. C. Hart, Jr., "AmorphousCarbon Hemetically Coated Optical Fibers," in Technical Digest,Optical Fiber Communication Conference (Optical Society ofAmerica, Washington, DC, 1988), paper TOG2.

5. R. Tuminaro, "Materials Aspects of the SL Lightguide UnderseaCable Design," Matls. Res. Soc. Bull. 13, 14-22 (1988).

6. E. Suhir, "Effect of Initial Curvature on Low Temperature Mi-crobending in Optical Fibers," IEEE/OSA J. Lightwave Technol.6, 1321-1327 (1988).

7. S. P. Timoshenko, Theory of Elasticity (Van Nostrand Reinhold,New York, 1989).

8. E. Suhir and T. M. Sullivan, "Analysis of Interfacial ThermalStresses and Adhesive Strength of Bi-Annular Cylinders," Int. J.Solids Struct. v. 26, No. 5/6, 581-600 (1990).

9. J. T. Krause (1989), private communications.

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